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It uses a strange orthography I haven't encountered before. Everything that is not in kanji is written in katakana, including particles, inflections, etc.

Is this the normal way, or one of the normal ways, of writing Japanese at that time?

The book is a handbook of the Okinawan language for Japanese speakers. All the Okinawan is written in katakana while all the Japanese is written in Kanji plus katakana. So it could be that the author of the handbook made a stylistic choice because of the type of book and didn't use the normal Japanese orthography of the day.

歴史的仮名遣とは ... 明治から第二次世界大戦終結直後までの公文書や学校教育において用いられたものであり、平安時代初期までの発音を反映した表記であると仮想されたものを基点としている。
The Historical Kana Orthography was used in pre-World War II official documents and schools.

平仮名に比べ学問的傾向が強いので、戦前の日本ではより正式な文字とみなされ、法令全書その他の公文書で用いられ、教育面でも平仮名に先行して教えられた。
Before World War II, Katakana was considered more formal and used in official documents. It was taught before hiragana in schools.